Condition on u so that | x ⋅ u u ⋅ u u − x | &lt; | x | If u , x ∈ R n then am I right to assert that the above inequality will hold if u makes an angle α ∈ ( 0 , π / 2 ) with span { x } ⊥ ?When n=3 it seems quite apparent that | x ⋅ u u ⋅ u u − x | = cos ⁡ ( α ) | x | &lt; | x | . I think it's exactly the same in higher dimensions.

Question

Condition on u so that       |              x      ⋅      u              u      ⋅      u        u  −  x      |    &amp;lt;      |    x      |  If   u  ,  x  ∈            R        n    then am I right to assert that the above inequality will hold if u makes an angle   α  ∈  (  0  ,  π      /    2  ) with span   {  x      }          ⊥      ?When n=3 it seems quite apparent that       |              x      ⋅      u              u      ⋅      u        u  −  x      |    =  cos  ⁡  (  α  )      |    x      |    &amp;lt;      |    x      |    .   I think it&#039;s exactly the same in higher dimensions.

levraijournalk1o · Accepted Answer

The expression only makes sense when   u  ≠  0, and then             u      ^        :=            x      ⋅      u              u      ⋅      u        u is the projection of x  onto u, i.e.             u      ^       is parallel to u and   x  =  v  +            u      ^       such that   v  ⋅  u  =  v  ⋅            u      ^        =  0Consequently,       |              x      ⋅      u              u      ⋅      u        u  −  x      |    =      |    v      |    &amp;lt;      |              u      ^        +  v      |    =      |    x      |   for all x

If u,x in RR^n then am I right to assert that the above inequality will hold if u makes an angle alpha in (0,pi/2) with span{x}^(_|_)?

Answered question

Answer & Explanation

New Questions in Precalculus